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8

The conductance of superconducting-normal hybrid structures

O. Entin-Wohlman,1, 2, ∗ Y. Imry,3 and A. Aharony1, †

1Department of Physics, Ben Gurion University, Beer Sheva 84105, Israel2Albert Einstein Minerva Center for Theoretical Physics,

Weizmann Institute of Science, Rehovot 76100, Israel3Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel

(Dated: October 25, 2018)

The dc conductance of normal-superconducting hybrid structures is discussed. It is shown thatsince the Bogoliubov-DeGennes (BDG) equation does not conserve charge, its application to createa Landauer-type approach for the conductance of the NSN system is problematic. We ‘mend’ thisdeficiency by calculating the conductance from the Kubo formula for a ring configuration (for thisgeometry the solutions of the BDG equation conserve charge). We show that the presence of asuperconductor segment within an otherwise normal metal may reduce the overall conductance ofthe composite structure. This reduction enhances the tendency of the NS composite to becomeinsulating.

PACS numbers: 74.45.+c,73.40.-c

Keywords: frequency-dependent mesoscopic conductance, superconducting-normal junctions

I. INTRODUCTION

In a seminal paper published a long time ago, Blonder,Tinkham, and Klapwijk1 (BTK) calculated the conduc-tance of a normal (N)–superconducting (S) interface asa function of the interface transparency. In particularthey showed that at zero temperature (T = 0) and forelectrons at the Fermi energy that conductance is givenby

GNS =2e2

h

(

1− |See|2 + |She|2)

, (1)

where |See|2 is the “normal” reflection coefficient of theinterface, while |She|2 is the reflection coefficient for theAndreev processes,2 which reflect electron-like excita-tions as hole-like ones. For simplicity, both the nor-mal and the superconductor regions were taken to befree of any impurity scattering (except at the interface).Then, when the interface is perfectly transparent, i.e.,|See|2 = 0 and |She|2 = 1, the value of GNS is twicethat of the “normal” quantum limit of the conductance.However, when a large enough barrier exists at the in-terface between the superconducting and the normal re-gions, GNS becomes much smaller than the conductanceobtained when the superconductor is made normal. Thisis the simplest example where superconductivity in apart of a system reduces its overall conductance. (Themore complicated many-channel, disordered case is notaddressed in this paper.) In this paper we consider themore subtle NSN combination, and demonstrate a simi-lar effect: with a large enough barrier at even one of theNS interfaces, the appearance of superconductivity in theS region reduces the overall conductance.Some of the motivation for the present work comes

from our wish to understand why the mixed NS bulkcomposite structure is often insulating at T = 0 and thesuperconducting phase of a thin film goes over (with in-creasing disorder or decreasing film thickness) directly

into the insulating rather than to the normal-conductingphase.3,4 It is interesting that often the activation energyfor the conductance of the insulating phase is given by thesuperconductor gap of the superconducting component.4

Thus, the superconducting component plays the role ofan additional barrier between the normal segments! Thecharge-vortex duality5 for a system of charged bosons ofcourse explains the insulating phase as dual to the super-conducting one. Vortex localization yields zero resistanceand charge localization yields zero conductance at T = 0.Our purpose is to provide a heuristic understanding ofhow the charge localization is established. Reducing thesmall-scale conductance of the system pushes it towardsthe insulating state. The NSN system is the simplest mi-croscopic element of the NS network. We find that it al-ready presents nontrivial theoretical questions having todo with a seeming deficiency of the Bogoliubov-DeGennes(BDG) formulation.

Blonder et al.1 employed the BDG equation for thequasiparticle excitations in the superconducting region,assuming that the energy gap ∆ which vanishes in thenormal part does not vary spatially in the superconduc-tor. Since in most situations the superconducting coher-ence length ξ is much larger than the Fermi wavelengthand much smaller than the length of the S-region, thisassumption seems quite harmless. Consequently, it iswidely accepted that the use of the BDG equation, with-out attempting to compute the (complex, in general) su-perconducting order-parameter self-consistently, is validfor many hybrid structures.

A particularly important issue emphasized by BTKconcerns the conversion of the normal current intoa super-current at the NS interface, ensuring charge-current conservation over the entire structure (see alsoRef. 6). Blonder et al.1 showed that the normal charge-current (which is distinct from the quasiparticle current)entering the superconductor decays, but concomitantlya super-current grows up gradually, until very far inside

2

S the normal current disappears completely, and the en-tire charge is carried away by Cooper pairs. One mightwonder what happens to this scenario when the super-conductor has a finite width and is not infinite as in theBTK case. It turns out that within the BDG formulationthe super-current in the S region does not properly con-vert back to normal current in the second N region. Weare not aware of a way to correct for this deficiency (leav-ing aside the possibility mentioned above to compute theorder-parameter self-consistently). In this paper we willcircumvent this problem by using a particular geometry.Not surprisingly, transport through hybrid NS struc-

tures has been addressed before, even prior7 to the pub-lication of the paper by BTK. Indeed, as Eq. (1) bears astrong resemblance to the Landauer formula8 for coher-ent transport, several modern treatments (representativeexamples are to be found in Refs. 9,10,11,12) employscattering theory within the Landauer picture in an at-tempt to extend the BTK result (1) to more than a sin-gle NS interface. The scattering formalism is also usedto study the current fluctuations in mesoscopic systemswith Andreev reflections.11,12 However, there is a viciouscaveat in this approach: the BDG equation, while con-serving the number of quasiparticles, does not conservecharge. The BDG formulation follows, in this respect,the similar deficiency of the BCS approach upon whichthe BDG equation is based. This does not cause anyharm when the formulation is used for a bulk supercon-ductor, as had been done by BTK (see also Ref. 13), orwhen the system has the shape of a closed ring, wherethe boundary conditions enforce current conservation.14

However, when the size of the superconducting segmentis finite, the non-conservation of the charge leads to cur-rent vs. chemical-potential-difference relations (withinthe linear response regime) that depend on the chemicalpotential of the superconductor, as opposed to the situ-ation in the normal phase.9,10 Indeed, the conductancesof various hybrid structures have been determined withinscattering theory (when the superconducting is “float-ing”) by fixing the chemical potential on S such that cur-rent is conserved.9,10 One would have assumed naıvelythat this procedure cures the problem mentioned above.It turns out, however, that it does not: as discussed byAnantram and Datta11 (see also Ref. 12), applying thevery same procedure to the calculation of the current fluc-tuations (the power spectrum of the noise) violates theJohnson-Nyquist relationships. We show in Sec. V thatthis ‘floating-potential’ approach also does not producethe doubling of the conductance [see Eq. (1) above] whenthe NS interface becomes transparent. It approaches,however, the conductance found by the Kubo formula inthe limit where the barrier is strong and its transparencyis low.It follows that in order to use the BDG equation for

the calculation of the conductance of a mesoscopic systemcontaining a finite-length superconducting segment, oneneeds charge-conserving solutions of that equation. Suchsolutions arise naturally when the NSN structure (theleft panel in Fig. 1) is closed to form a “ring” (the rightpanel in Fig. 1). One can use those solutions in the Kuboformula for the conductivity of a large system. This isthe approach adopted in this paper.

SN N'

ce+

ce-

ch-

ch+

ch-

ch+

ce-

ce+

dS

S

N

dS

dN

ce+

ce-

ch+

ch-

ce+ce

-ch-ch

+

FIG. 1: Left panel: the ‘open’ NSN structure, where the perpendicular lines represent potential barriers (the explicit modeltreated in Sec. IV assumes that the right interface is clean). Right panel: the corresponding ring NSN system. The incomingand outgoing amplitudes of the electron-like and the hole-like waves on both sides of the superconductor are marked by arrows.

We address the simplest problem of a one-dimensionalNSN structure at zero temperature (we do not discuss

here the complications arising in the multi-channel caseincluding disorder). After discussing the conductance be-

3

tween the two N segments (left panel in Fig. 1), and deal-ing with the nontrivial problem arising from the above-mentioned deficiency of the BDG picture, we find thatthe NSN conductance can indeed decrease when, for ex-ample, the superconducting component becomes longer.We are not able to treat here the behavior of the two-dimensional or the three-dimensional NSN arrays. Suf-fice it to say that when the “small scale” resistance of theelementary building block increases, the tendency for lo-calization at larger scales becomes stronger.

II. DIFFICULTIES WITH THE

LANDAUER-TYPE FORMULATION FOR THE

NSN CONDUCTANCE

The subtleties involved in producing a consistentLandauer-type formula for the conductance of an NSNstructure within the BDG formulation are best explainedby considering the simplest single-mode, two-terminalconfiguration at zero temperature. In other words, weassume that there is no scattering in the system exceptfor the potential barriers at the interfaces, such that thetransverse channel modes are not mixed (and their in-dices can be omitted).We start with the purely normal case, as shown in the

left panel of Fig. 1, except that the S section is replacedby a normal one (for example, by letting its gap approachzero). In this case there is no need to treat electrons andholes concomitantly, as there are no Andreev processes.It is enough to use only electrons (or only holes). We de-note the reflection coefficient for an electron coming fromthe left by R, and the one for an electron coming fromthe right by R′. Likewise, the transmission coefficientfrom the left to the right is denoted by T , and the onefrom the right to the left is T ′. Unitarity (particle con-servation, which is also charge conservation in this case)implies

R+ T = R′ + T ′ = 1 . (2)

Time-reversal symmetry implies further that T = T ′,and hence R = R′. Next we assign to the left conductora chemical potential µL and to the right one a chemicalpotential µR. In the linear response regime, µL−µR → 0.

The middle conductor, of a finite length, is kept floating,(i.e., it is not connected to any reservoir), and will ac-quire a chemical potential µn. Clearly, the right-goingcurrents to the left of the middle segment, IL, and to itsright, IR, are given by15

IL =2e

h

(

(1 −R)(µL − µn)− T ′(µR − µn))

,

IR =2e

h

(

−(1−R′)(µR − µn) + T (µL − µn))

. (3)

From the unitarity condition (2) and time-reversal sym-metry, it follows that IL = IR ≡ I = (2e/h)T (µL − µR),independently of the value of µn (which, in fact, drops outof the two equations). The well-known Landauer formulafor the conductance,

G = (2e2/h)T , (4)

is immediately obtained, and is independent of µn, as itshould be.16

When the middle section is a superconductor, furtherAndreev-type processes become possible. An electroncan be reflected/transmitted as a hole, and vice versa.

For an electron incident from the left, the probabilitiesfor the Andreev reflection and transmission processes aredenoted RA and TA, respectively. The correspondingquantities for an electron coming from the right are R′

A

and T ′A. The unitary condition (2) is now replaced by

R+ T +RA + TA = R′ + T ′ +R′A + T ′

A = 1 . (5)

However, the charge conservation condition now reads17

R+ T −RA − TA = R′ + T ′ −R′A − T ′

A = 1 . (6)

The two conditions, Eqs. (5) and (6), are not compatiblewhenever the Andreev probabilities do not vanish, exceptfor a ring geometry (where their consistency is enforced).Moreover, while Eq. (5) always holds for the solutions ofthe BDG equation, Eq. (6) does not!

The expressions for the currents [see Eqs. (3)] nowbecome (note that group velocity of the holes is oppositeto that of the electrons)

IL =2e

h

(

(1−R+RA)(µL − µS)− (T ′ − T ′A)(µR − µS)

)

,

IR =2e

h

(

−(1−R′ +R′A)(µR − µS) + (T − TA)(µL − µS)

)

, (7)

where µS is the chemical potential on the superconduc-tor. We note that because charge conservation does nothold [see Eq. (6)], µS does not drop out of these equa-

tions. Its value is relevant. Equations (7) are of the sameform as Eqs. (2) of Takane and Ebisawa:9 these authorsdetermine µS so that IL = IR. It is then possible to

4

obtain a conductance from the current-to voltage ratio,as was done in Ref. 9. We reproduce their result for theNSN conductance in Sec. V, see Eq. (57) there. It isdisturbing, however, that the determined value of µS isrelevant. Moreover, for the result for the conductance[Eq. (57)] to be fully satisfactory, it should agree withthe linear response, Kubo, formula for the related largering geometry. This is the case for the normal Landauerformula, but, in general not for Eq. (57) (see Sec. V formore details).The above considerations, in particular, Eqs. (5) and

(6), can be put on a more general basis. By imposingthe appropriate boundary conditions on the plane-wavesolutions of the BDG equation it is possible to derive thescattering matrix, S, of the NSN structure. This (4×4)matrix relates the amplitudes of the incoming waves tothose of the outgoing ones, (see left panel in Fig. 1)

cout = Scin . (8)

Here,13 the incoming amplitudes are

cin =(

c+e (N), c−e (N′), c−h (N), c+h (N

′))

, (9)

and the outgoing ones are

cout =(

c−e (N), c+e (N′), c+h (N), c−h (N

′))

. (10)

In Eqs. (9) and (10), c+e,h(N) denotes the amplitude

of an electron-like (hole-like) excitation with a positivewave vector ke,h incident from the left normal side whilec−e,h(N) refers to the waves having negative wave vectors.Since the BDG equation conserves the number of quasi-particles, the scattering matrix S is necessarily unitary,and therefore

c†outcout = c†incin . (11)

However, conservation of the charge current1,17 requires[see Eq. (6)]

|c+e (N )|2 − |c−e (N )|2 + |c+h (N )|2 − |c−h (N )|2

= |c+e (N ′)|2 − |c−e (N ′)|2 + |c+h (N ′)|2 − |c−h (N ′)|2 .(12)

Comparing Eqs. (11) and (12), we see that they imply

|c+e (N )|2 + |c−e (N ′)|2 = |c+e (N ′)|2 + |c−e (N)|2 ,

|c−h (N)|2 + |c+h (N ′)|2 = |c+h (N)|2 + |c−h (N ′)|2 . (13)

Namely, the sum of the amplitudes squared of the incom-ing electron-like excitations is equal to the sum of the am-plitudes squared of the outgoing electron-like excitations,and so is the situation for the hole-like ones. These condi-tions are the same as those that would have been derivedfrom Eqs. (5) and (6), had we required that both condi-tions should be satisfied together. This always holds fora normal system, in which these two types of quasipar-ticles are not mixed. However, in a superconductor the

Andreev processes mix the hole-like with the electron-like excitations, thus violating the conditions (13) forgeneral hybrid structures with a finite-size S segment.An exception is the ring geometry. There, (see Fig. 1)the ratios c+e,h(N)/c+e,h(N

′) are necessarily phase factors,

and so are the ratios c−e,h(N)/c−e,h(N′). As a result, the

plane-wave solutions of the BDG equations for the ringgeometry do satisfy both conditions (13), namely, thesesolutions conserve charge. One may therefore employthe BDG equation for the ring geometry in the contextof the Kubo formulation to calculate the conductance ofthe NSN structure.

III. THE KUBO FORMULA FOR A LARGE

RING

For an infinite system, the Kubo-type conductivity atfrequency ω may be most easily obtained by calculat-ing, using the golden rule, the power absorbed by thesystem from a classical monochromatic electromagneticfield. We consider for simplicity noninteracting fermions(or Fermi quasiparticles), and focus on the σxx compo-nent of the conductivity,

σxx(ω) = −πe2

V

1

ω

∑

j,ℓ

|〈j|vx|ℓ〉|2

× δ(ǫℓ − ǫj − ~ω)(f(ǫj)− f(ǫℓ)) . (14)

Here, |j〉 and |ℓ〉 are the quasi-electron states and f(ǫj)and f(ǫℓ) their populations. In Eq. (14), V is the volumeof the system, to be sent to infinity at the end of the cal-culation, at which stage the summations over the statesare replaced by integrations with the densities of states.The x−component of the velocity operator is denoted vx.For the case of a normal (i.e., non superconducting) scat-terer with infinite leads the equivalence of the Kubo andthe two-terminal Landauer approaches has been estab-lished in Refs. 18 and 19.The assumption of an infinite system is crucial in or-

der to have a continuum of states. An isolated finitesystem with a truly discrete spectrum does not in factabsorb energy from the monochromatic field. In orderto obtain a finite conductivity for a finite large system,it has to be (and it is, in most real situations) coupledto a very large heat bath. For example, to an assemblyof thermal phonons. This enables energy to be trans-ferred from the electromagnetic field into the bath viathe small electronic system. For a weak enough interac-tion with the bath, one may say that the discrete levels ofthe system have acquired finite widths, ηj . It then makessense to write down Eq. (14) with the levels having a fi-nite width (or with an imaginary part to the frequencyω, which will amount to a non-monochromatic drivingfield). This procedure has been discussed, including thedc limit (Reω → 0), by Thouless and Kirkpatrick,20 fol-lowing Czycholl and Kramer,21 and used for example inRef. 22, see also Ref. 15. It is postulated, and can

5

be demonstrated in typical cases, that once the ηj ’s arelarger than the level spacing near the Fermi energy, butmuch smaller than all other relevant energy scales in theproblem, this procedure yields the physically relevantlow-frequency conductance of the system.It hence follows that the ω → 0 conductance is ob-

tained upon transforming the summations in Eq. (14)into energy integrations. This allows one to approxi-mate f(ǫℓ + ~ω) − f(ǫℓ) ≃ ~ωf ′(ǫℓ). Focusing on ourone-dimensional configuration, we take x along the ringcircumference, and replace the volume of the system byits length, d. Since the (one-dimensional) conductance isrelated to the conductivity by G ≡ σ/d, we recover theKubo-Greenwood-type formula at low frequencies

G =e2h

2ν2∑

deg

|〈|v|〉|2 , (15)

where the sum is over the (almost) degenerate initialstates and over the (almost) degenerate final states,within the narrow range ~ω (→ 0) above those initialstates, and all states are at about the Fermi energy. Onemight also say that the cancellation of the frequency [seeEq. (14)] is caused by the fact that the initial state was,at T = 0, within ~ω of the Fermi level. In Eq. (15),the matrix element squared of the velocity was replacedby its typical value in the small relevant energy windowaround the Fermi energy. The double sum of Eq. (14)gave rise to two factors of the single-particle density ofstates (per unit energy, per unit length, and per spin), ν,

ν = 1/(hvF) , (16)

in the one-dimensional system. Comparing Eq. (15) withthe “traditional” Landauer formula (4), we find that inthe Kubo approach the total transmission is replaced bythe appropriate sum over the velocity matrix elementssquared, i.e.,

G =2e2

h

1

4v2F

∑

deg

|〈|v|〉|2 . (17)

It is instructive to review the way the Kubo formula inits form (17) produces the Landauer result for the usualtwo-probe geometry. We consider initial left-going scat-tering states. These are degenerate with the right-goingones. This degeneracy gives a factor of two in the finalresult, to which the spin degeneracy adds another fac-tor of two. Each such state will have a matrix element,(1−R+ T )/2 = T , with the appropriate final left-goingscattering state, and rt with the final right-going scat-tering state (r and t are the reflection and transmissionamplitudes, respectively). Adding the absolute valuessquared together yields T (note the cancellation of the T 2

term!). Introducing the above degeneracy factors gives∑

deg |〈|v|〉|2 = 4v2FT , and thus reproduces the Landauer

result, Eq. (4) above.For the ring geometry the states are stationary and

normalizable. Taking as a representative example a nor-mal ring with a single delta-function potential, one finds

that the ratio of the amplitudes of the clockwise mov-ing wave and anticlockwise moving one is a phase factor,exp[iφe], where on the Fermi energy exp[iφe] = −1 or(1 − iζ)/(1 + iζ). Here ζ is the strength of the deltafunction potential, with the corresponding transmissionT = 1/(1 + ζ2). The velocity matrix elements are thenvF/(1 ± iζ), and thus together with the spin degener-acy reproduce the Landauer formula (4). We give moredetails in the next section, which is devoted to the eval-uation of the states and the current matrix elements foran NS ring, and the case of an entirely normal ring istreated as a limiting case.

IV. THE VELOCITY MATRIX ELEMENTS

Here we compute the matrix elements of the velocityoperator, which are used in the Kubo formula (17) for theconductance. We follow BTK in assuming that the entirescattering takes place only at the NS interfaces, and thatthe pair-potential ∆ is finite and spatially-invariant inthe superconducting region, and vanishes in the normalone. Since then there is no channel mixing, the problembecomes effectively one-dimensional, and the quasipar-ticles are described by the one-dimensional Bogoliubov-DeGennes equation, (we use in this section units in which~ = 1)[

− 12m

d2

dx2 − EF ∆

∆ 12m

d2

dx2 + EF

]

Ψ(x) = ǫΨ(x) . (18)

Note that this equation takes into account the two possi-ble spin directions pertaining to a certain energy ǫ (mea-sured from the Fermi level). In N, where ∆ = 0, thesolutions of Eq. (18) are

Ψ±e (x) =

[

10

]

e±ikex , Ψ±h (x) =

[

01

]

e±ikhx , (19)

with the wave vectors

ke,h =√

2m(EF ± ǫ) ≃ kF ± ǫ/vF . (20)

In the superconducting segment the solutions are

Ψ±e (x) =

[

uv

]

e±iqex , Ψ±h (x) =

[

vu

]

e±iqhx . (21)

Here,

qe,h =√

2m(EF ± Ω) ≃ kF ± Ω/vF , (22)

where

Ω =√

ǫ2 −∆2 , ǫ ≥ ∆ ,

Ω = i√

∆2 − ǫ2 , ǫ ≤ ∆ , (23)

and

u2 =ǫ

Ω

(1

2+

Ω

2ǫ

)

, v2 =ǫ

Ω

(1

2− Ω

2ǫ

)

. (24)

6

The factor [ǫ/Ω]1/2 compensates for the different groupvelocity of the quasiparticles in the superconductor[∂ǫ/∂q = (q/m)(Ω/ǫ)], and it multiplies the usual coher-ence factors u and v, u = [ǫ/Ω]1/2u and v = [ǫ/Ω]1/2v.The amplitudes c±e,h(N) (see Fig. 1) are the coefficients

of the waves exp[±ike,hx]. Analogous amplitudes are de-fined for the waves in the superconducting segment.

For simplicity, we assume that the (left) NS interface atx = 0 (see Fig. 1) is represented by a delta-function po-tential, λδ(x), of strength ζ = λ/vF. Then the boundaryconditions17 are the continuity of the wave functions andthe discontinuity (of magnitude ζ) of their derivatives,leading to the following relations among the amplitudesof the N region and those of the S one,

c+e (S)c−e (S)c+h (S)c−h (S)

=

u 0 −v 00 u 0 −v−v 0 u 00 −v 0 u

1− iζ −iζ 0 0iζ 1 + iζ 0 00 0 1− iζ −iζ0 0 iζ 1 + iζ

c+e (N)c−e (N)c+h (N)c−h (N)

. (25)

The other NS interface, located at x = dS , is assumed to be perfectly transparent, and then the boundary conditionsare the continuity of the wave functions and their derivatives. When the system has the shape of a ring, in which thelength of the normal segment is dN , these boundary conditions are

e−ikFdNγ−1N c+e (N)

eikFdNγNc−e (N)e−ikFdNγNc+h (N)eikFdNγ−1

N c−h (N)

=

eikFdS uγS 0 eikFdSγ−1S v 0

0 e−ikFdS uγ−1S 0 e−ikFdSγS v

eikFdS vγS 0 eikFdSγ−1S u 0

0 e−ikFdS vγ−1S 0 e−ikFdSγS u

c+e (S)c−e (S)c+h (S)c−h (S)

, (26)

where

γN = eiǫdN/vF , γS = eiΩdS/vF . (27)

Without loss of generality we may choose exp(ikFd) = 1,where d = dN + dS is the total length of the ring. ThenkF disappears from the boundary conditions.Upon eliminating the S-region amplitudes, one obtains

the equation which determines the allowed eigenenergiesof the ring, and the ratios among the amplitudes of thenormal region for each such energy,

(

X 0 Y 00 X∗ 0 Y ∗

Y ∗ 0 X∗ 00 Y 0 X

−

1− iζ −iζ 0 0iζ 1 + iζ 0 00 0 1− iζ −iζ0 0 iζ 1 + iζ

)

c+e (N)c−e (N)c+h (N)c−h (N)

= 0 .

(28)

Here,

X = γ−1N

(

cos(ΩdS/vF)− iǫ

Ωsin(ΩdS/vF)

)

,

Y = 2iγN uv sin(ΩdS/vF) , (29)

such that |X |2−|Y |2 = 1 for both ǫ ≥ ∆ and ǫ ≤ ∆. Theallowed eigenenergies are given by the vanishing of thedeterminant of the matrix in Eq. (28). The zeroes of thedeterminant define the families of possible eigenenergies,

which are rather dense when the size of the entire systemis large. When ζ 6= 0, the eigenvectors of the matrix(28) are such that the ratios of the clockwise electron(hole) waves to the anticlockwise electron (hole) ones (seeFig. 1) for each of the families of eigenenergies are phasefactors,

c−e (N)

c+e (N)= eiφe ,

c−h (N)

c+h (N)= eiφh . (30)

In other words, at any finite value ζ of the barrier, thereis a perfect reflection of the electron and the hole waves(in the ring geometry). On the other hand, the ratios ofthe hole amplitudes to the electron ones obey

c+h (N)

c+e (N)= P ,

c−h (N)

c−e (N)= P ∗ , (31)

such that the phase of P is (φe − φh)/2,

P = |P |ei(φe−φh)/2 . (32)

It is illuminating to consider Eq. (28) and its solu-tions in the limit of very high energies, ǫ ≫ ∆, wherethe superconducting order parameter ∆ becomes irrele-vant, and the entire system behaves as if it were normal.Then [see Eqs. (29)] X = exp[iǫd/vF] and Y = 0, andEq. (28) separates into two independent blocks, for theelectron-like excitations, and for the hole-like ones. The

7

eigenenergies are given by

eiǫd/vF = 1 or1 + iζ

1− iζ, for the electron waves ,

eiǫd/vF = 1 or1− iζ

1 + iζ, for the hole waves , (33)

with the corresponding phase ratios

eiφe = eiφh = −1 , or1− iζ

1 + iζ. (34)

(In this limit P , the ratio of the hole amplitude to theelectron amplitude, is not defined.) We show below thatthese are the phase factors exp[iφe,h] which determine theconductance (4) when calculated from the Kubo formula(17).In the other extreme limit of sub-gap energies, ǫ ≪ ∆,

one approximates1 [see Eq. (23)]

Ω

vF≃ ∆

vF≡ 1

ξ, (35)

and consequently [see Eqs. (29)]

X ≃ γ−1N cosh(dS/ξ) , Y ≃ −iγNsinh(dS/ξ) , (36)

where ξ is the coherence length in the superconductor.One then finds a quadratic equation for cos(ǫdN/vF). Wedo not present explicit expressions for the solutions andthe amplitude ratios since they are rather cumbersome.The next step in this calculation is to find the normal-

ization of the wave functions, using∫ dS

0

dx|ΨS(x)|2 +∫ d

dS

dx|ΨN (x)|2 = 1 . (37)

In the N region

|ΨN (x)|2 = |c+e (N)|2 + |c−e (N)|2

+ (e2ikF xc−e (N))∗c+e (N) + cc) + (e → h) . (38)

When dN is large, such that the oscillatory terms (inkFdN ) can be ignored, the contribution of the normalpart to the normalization integral becomes

∫ d

dS

dx|ΨN (x)|2 = dN

(

|c+e (N)|2 + |c−e (N)|2

+ |c+h (N)|2 + |c−h (N)|2)

. (39)

The calculation of the contribution to the normalizationcoming from the S region is more subtle, since the wavevectors can have an imaginary part [see Eqs. (22) and(23)]. Disregarding terms oscillating with kFdS , we find

|ΨS(x)|2 → (|u|2 + |v|2)(

(|c+e (S)|2 + |c−h (S)|2)ei (Ω−Ω∗)x

vF

+ (|c−e (S)|2 + |c+h (S)|2)ei (Ω

∗−Ω)xvF

)

+ (u∗v + uv∗)(

ei(Ω+Ω∗)x

vF ((c−e (S))∗c−h (S)

+ (c+h (S))∗c+e (S)) + cc

)

. (40)

This rather complicated result reflects the fact (specif-ically, its second part) that in the superconductor theelectron waves are mixed with the hole ones. However,at very large energies, ǫ ≫ ∆, or at very small ones,ǫ ≪ ∆, the mixing term, (u∗v + uv∗), vanishes. In thefollowing, we confine ourselves to these two limits. Inthe high-energies limit the normalization of either theclockwise waves or the anticlockwise ones is simply

√2d,

where d is the total length. In the low-energies limit wefind

∫ dS

0

dx|ΨS(x)|2 =ξ

2sinh(dS/ξ)(|c+e (N)|2 + |c−e (N)|2)

×(

edS/ξ|Ma|2 + e−dS/ξ|Mb|2)

, (41)

where

Ma = γ−1N − iγNP ,

Mb = γ−1N + iγNP , (42)

and P is given by Eq. (31).Having fully determined the wave functions, it remains

to compute the matrix elements of the velocity,

vjℓ ≡ 〈j|v|ℓ〉 = 1

2mi

∫ d

0

dx(

Ψ∗j

dΨℓ

dx−Ψℓ

dΨ∗j

dx

)

, (43)

with the indices j and ℓ enumerating the various eigen-functions. As in the calculation of the normalization,here again there are contributions from the normal andfrom the superconducting regions. In each region we dis-card the oscillatory terms, those which involve kFdN orkFdS .The contribution of the normal part to the integral in

Eq. (43) reads

vNjℓ =dNkFm

(

(c+e (N))∗j (c+e (N))ℓ − (c−e (N))∗j (c

−e (N))ℓ

+ (c+h (N))∗j (c+h (N))ℓ − (c−h (N))∗j (c

−h (N))ℓ

)

. (44)

In the high-energies limit, ǫ ≫ ∆, the contribution of thesuperconducting segment to the integration is the sameas (44) (with the arguments N replaced by S, and dNreplaced by dS). The contribution of the S part in thelimit of very low energies, ǫ ≪ ∆, is

vSjℓ =ξkF2m

sinh(dS/ξ)

×(

(c+e (N))∗j (c+e (N))ℓMjℓ − (c−e (N))∗j (c

−e (N))ℓM

∗jℓ

)

,

(45)

where we have denoted [see Eqs. (42)]

Mjℓ = edS/ξ(Ma)∗j (Ma)ℓ + e−dS/ξ(Mb)

∗j (Mb)ℓ . (46)

It is again useful to examine the limit of high energies,ǫ ≫ ∆, where the entire ring behaves as if it were normal.Then, the electron- and the hole-like waves are separated.

8

The spectrum and the amplitude ratios are given by Eqs.(33) and (34), and the normalization for each species is√2d. The matrix elements of the velocity are simply

velejℓ =kFd

m(c+e )

∗j (c

+e )ℓ(1− ei(φ

ℓe−φj

e)) , (47)

and an analogous result is obtained for the contributionof the hole waves. Obviously, the diagonal ones vanish.The non-diagonal ones give (kF/m)/(1± iζ), and conse-quently

∑

deg

|〈|v|〉|2 = 4v2FT , T =1

1 + ζ2. (48)

Note that the non-vanishing matrix elements arise fromthe phase factor between waves belonging to the samespecies but moving along opposite directions. Hence, theKubo formulation for the ring geometry reproduces theLandauer result for the dc conductance.Another illuminating limit is when ζ vanishes, and

both NS interfaces (see Fig. 1) are perfectly transparent,in which case the clockwise and the anticlockwise ampli-tudes are independent. The matrix elements of the ve-locity, for sub-gap energies, are (for either the clockwise-moving or the anticlockwise-moving excitations)

vℓℓ = vF ,

vjℓ = v∗ℓj = vFdN

dN + ξtanh(dS/ξ)

× sinh(dS/ξ)[sinh(dS/ξ) + i]

cosh2(dS/ξ). (49)

(It is interesting to note that the off-diagonal matrix el-ements are coming from the N region alone.) Hence thecontribution of both the clockwise waves and the anti-clockwise waves is

∑

deg

|〈|v|〉|2 = 4v2F

(

1+[ dN tanh(dS/ξ)

dN + ξtanh(dS/ξ)

]2)

. (50)

Thus, when dS/ξ tends to zero (namely, in the absence ofthe superconductor) the result approaches the Landauerformula for a transparent barrier, cf. Eq. (48). On theother hand, when dN ≥ dS ≫ ξ, our result (50) tends tothe one found by BTK1 (for a clean interface), namely,it is twice the value of the quantum conductance.

Unfortunately, the explicit expressions for the velocitymatrix elements at low energies for general values of ζ anddS/ξ are rather complicated. Consequently, we presentthe results of the calculations only graphically, see Figs.2 and 3. The figures show the conductance [divided by2e2/h, see Eq. (17)] as a function of the ratio dS/ξ forvarious values of the interface transmission, T = 1/(1 +ζ2), and as a function of that transmission, for variousvalues of the size of the superconductor segment, dS/ξ.

0 2 4 6 8 10 12 14dS Ξ

0.5

1.0

1.5

2.0GH2e2hL

dN=dS

0.0 0.2 0.4 0.6 0.8 1.0T

0.5

1.0

1.5

2.0GH2e2hL

dN=dS

FIG. 2: Left panel: Conductance vs dS/ξ for several values of the transmission. Right panel: Conductance vs the transmissionfor several values of dS/ξ. Here the length of the S region equals that of the N region (dN = dS). All information pertainingto values of dS smaller than ξ is presented by dashed curves.

Figure 2 presents the results for the case dN = dS , i.e.,the segments N and S of the ring are of equal lengths.In the left panel the conductance is plotted as a function

of dS/ξ, for values of T ranging between 1 (the upmostcurve) and 0.06, (the lowest-lying one). The main featureof these curves is the variation of their slope as the NS

9

barrier becomes less and less transparent. For T = 1 theconductance increases with the length of the supercon-ductor (until it is double that of the normal system inthe BTK limit where dS/ξ → ∞). As the transparencydecreases, the conductance, albeit increasing with dS/ξbecomes smaller, until at about T ≃ 0.8 it changes itsslope and begins deceasing as the size of the supercon-ductor is increased. The same characteristic behavior isobtained when the size of the normal part largely exceedsthat of the superconductor, as is depicted in the left panelof Fig. 3. The right panels in both Figs. 2 and 3 show

the (normalized) conductance as a function of the barriertransparency for various values of the superconductingsize, dS/ξ, ranging between 0.01 (almost a straight line)and 20 (parabolic curve). Here one observes that the con-ductance is linear in the barrier transmission as long asthe superconducting is small enough, and then becomesquadratic in T , for large values of dS/ξ. It should benoted, however, that the use of the BDG approach fordS/ξ ≪ 1 is dubious. For this region, we have presentedall information pertaining to such values by the dottedcurves.

0 2 4 6 8 10 12 14dS Ξ

0.5

1.0

1.5

2.0GH2e2hL

dN=9dS

0.0 0.2 0.4 0.6 0.8 1.0T

0.5

1.0

1.5

2.0GH2e2hL

dN=9dS

FIG. 3: Left panel: Conductance vs dS/ξ for several values of the transmission. Right panel: Conductance vs the transmissionfor several values of dS/ξ. Here the length of the S region is smaller than the length of the N region (dN/dS = 9). Allinformation pertaining to values of dS smaller than ξ is presented by dashed curves.

V. DISCUSSION

As is described in Secs. I and II, several previous cal-culations aiming to determine the conductance of hybridnormal-superconducting structures are based on the scat-tering matrix for the quasiparticles, as derived from theBDG equation.9,10,11 Our reservations regarding this pro-cedure are explained in Sec. II. Nonetheless, it is inter-esting to compare the conductance found from the Kuboformula and the one derived after fixing the chemical po-tential of the superconductor, as explained in Sec. II.Here we carry out this comparison for the model systemof Sec. IV.The scattering matrix of the NSN junction (see the

left panel in Fig. 1) is a function of the energy ǫ. Forour purposes here it suffices to derive it for zero energy,

i.e. on the Fermi level. This derivation is accomplishedby eliminating the amplitudes of the waves within thesuperconductor, using the boundary conditions (25), andthe boundary conditions at the (clean) interface betweenthe superconductor and the second normal layer, denotedN’ [note that when ǫ = 0, γN = 1, see Eq. (27)]

c+e (N′)

c−e (N′)

c+h (N′)

c−h (N′)

=

uγS 0 γ−1S v 0

0 uγ−1S 0 γS v

vγS 0 γ−1S u 0

0 vγ−1S 0 γS u

c+e (S)c−e (S)c+h (S)c−h (S)

.

(51)

As a result, the scattering matrix as defined in Eq. (8)takes the form

c−e (N)c+e (N

′)c+h (N)c−h (N

′)

=1

D

−iζ(1− iζ)(c2 + s2) (1− iζ)c isc −ζsc(1− iζ) −iζ(1− iζ) ζs isc(1 + 2ζ2)

isc ζs iζ(1 + iζ)(c2 + s2) c(1 + iζ)−ζs isc(1 + 2ζ2) c(1 + iζ) iζ(1 + iζ)

c+e (N)c−e (N

′)c−h (N)c+h (N

′)

, (52)

10

where

D = (1 + ζ2)c2 + ζ2s2 , (53)

and in order to shorten the notations we have denoted

s ≡ sinh(dS/ξ) , c ≡ cosh(dS/ξ) . (54)

Referring to the notations introduced in Sec. II, wefind from Eq. (52) that for an electron-like incident fromthe left

R =ζ2(1 + ζ2)(c2 + s2)2

D2, T =

c2(1 + ζ2)

D2,

RA =s2c2

D2, TA =

ζ2s2

D2, (55)

while for an electron-like wave coming from the right thecorresponding probabilities are

R′ =ζ2(1 + ζ2)

D2, T ′ =

c2(1 + ζ2)

D2,

R′A =

s2c2(1 + 2ζ2)2

D2, T ′

A =ζ2s2

D2. (56)

It is easy to verify that the conditions for quasiparticle-number conservation, Eq. (5), are obeyed by the proba-bilities (55) and (56), since the scattering matrix is uni-tary; Eqs. (6) for the charge conservation are not obeyed.Following Refs. 9 and 11, current conservation is now im-posed on Eqs. (7), leading to the determination of thechemical potential on the superconductor. This leads toa linear relation between IL and the chemical potentialdifference µL−µR, which is identified as the conductance.Denoting the latter by Gsc, one has

Gsc =gLLgRR − gLRgRL

gLL + gRR + gLR + gRL

, (57)

where9

gij =2e2

h

(

δij − |Seeij |2 + |She

ij |2)

. (58)

Here, i and j refer to the two sides of the junction, sayleft and right, and the superscripts ee or he refer to theparticular process. Thus for example, the 11 element ofthe matrix in Eq. (52) is See

LL, while the 41 element isSheRL.We compare the outcome of Eq. (57) with the conduc-

tance found from the Kubo formula in Fig. 4. There, theconductances are plotted for four values of the interfacetransmission, an almost perfect one, T = 0.96, (the up-most pair of curves), T = 0.8, (the second pair of curvesfrom above), T = 0.5 and T = 0.31, (the low-lying twopairs of curves). In each case, the result of Eq. (57)is the dashed line. There are three interesting featuresof this comparison. Firstly, the conductance found fromEq. (57) is always smaller than the one found from theKubo formula, Eq. (17). The difference between the tworesults decreases with increasing barrier (decreasing T )

and seems to vanish in the limit T → 0. Thus, while for anormal conductor the ring geometry and the simple two-terminal configuration produce identical results for theconductance, this is unfortunately no longer the case forthe NSN junction (NS for the ring geometry). The sec-ond interesting feature concerns the slopes of the curvesin Fig. 4, when the barrier transmission, T is varied.While the conductance computed from the Kubo formulashows a crossover of the slope, from being positive at high

values of T to being negative at lower values, the slope ofthe conductance found from Eq. (57) seems to be alwaysnegative. A third important difference between the twoapproaches is that for large ds/ξ and not-too-small T ,the Kubo result becomes larger than 2e2/h (tending to4e2/h in the limit ds/ξ → ∞ and T → 1), while Eq. (57)actually tends to GNS = 2e2/h and never yields the dou-bling of GNS due to the Andreev reflections. We blamethese differences between the two approaches on the lackof conservation of charge in the BDG formulation. Webelieve that this deficiency is corrected by employing theKubo formula for the ring geometry.

0 5 10 15 20 25 30dS Ξ

0.5

1.0

1.5

2.0GH2e2hL

FIG. 4: Comparison between the conductance Eq. (57)(dashed curves) and the conductance computed according tothe Kubo formula, Eq. (17) (solid lines), as a function ofdS/ξ for four values of the interface transmission. The latterconductance is always larger than the former.

The difference between the two approaches becomesmost marked in the limit of a nearly transparent barrierand a thick superconductor. This can be easily under-stood by noting that the addition of the two NS resis-tances is handled very differently by the two approaches.For the fully quantum case, adding two ideal conduc-tances (4e2/h for the NS case) gives just one ideal con-ductance. On the other, the scattering formalism, in thelimit ds/ξ → ∞ and T → 1 gives that both T (T ′) andTA (T ′

A) vanish, and so does R (R′), while RA (R′A)

tends to unity [see Eqs. (55) and (56)]. As a result,gLL = gRR = 4e2/h and gLR = gRL = 0, and Eq. (57)becomes exactly the classical addition of resistances, pro-ducing half the ideal quantum conductance of the pureNS junction. This is due to the fixing of the chemicalpotential on the S-section to conserve the current, as inthe classical treatment.

11

In summary, we have shown that the presence of a su-perconducting segment in an otherwise normal system re-duces the overall conductance once the barriers betweenthe superconducting and the normal parts become highenough. Thus, the superconducting segments may pushthe system towards the localizes insulating state.From the appearance of the plots presented in Figs. 2

and 3, one may be tempted to say that the system expe-riences a metal-insulator quantum phase transition froma finite to a vanishing conductivity at large ds/ξ, whenT decreases (which can be inferred to as “disorder in-crease”). We refrain here from making such a statementand defer the discussion of such a quantum phase tran-sition in the thermodynamic limit for the composite NS

system to future work.

Acknowledgments

We thank M. Schechter for many illuminating discus-sions. This work was supported by the German Fed-eral Ministry of Education and Research (BMBF) withinthe framework of the German-Israeli project cooperation(DIP), and by the Israel Science Foundation (ISF) andby the Converging Technologies Program of the IsraelScience Foundation (ISF), grant No 1783/07.

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